Answer:
c.![\frac{\pi}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpi%7D%7B2%7D)
Step-by-step explanation:
We are given that
![y=sin(x-C)](https://tex.z-dn.net/?f=y%3Dsin%28x-C%29)
We have to find the value of C for which given function is even function.
We know that
Even function : If f(x)=f(-x) then the function is called even function.
![a.2\pi](https://tex.z-dn.net/?f=a.2%5Cpi)
Substitute the value then we get
![y= sin(x-2\pi)= sin(-(2\pi-x))=-sin (2\pi-x)=sin x](https://tex.z-dn.net/?f=y%3D%20sin%28x-2%5Cpi%29%3D%20sin%28-%282%5Cpi-x%29%29%3D-sin%20%282%5Cpi-x%29%3Dsin%20x)
We know that sin (-x)=-sin x, ![sin(2\pi-x)=-sinx](https://tex.z-dn.net/?f=sin%282%5Cpi-x%29%3D-sinx)
We know that Sin x is an odd function , therefore, option a is incorrect.
b.
Substitute the value then we get
![y= sin (x-\pi)=sin(-(\pi-x))=-sin (\pi-x)=-sin x](https://tex.z-dn.net/?f=y%3D%20sin%20%28x-%5Cpi%29%3Dsin%28-%28%5Cpi-x%29%29%3D-sin%20%28%5Cpi-x%29%3D-sin%20x)
It is an odd function.
Hence, option b is incorrect.
c.![\frac{\pi}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpi%7D%7B2%7D)
Substitute the value then we get
![y= sin(x-\frac{\pi}{2})=sin(-(\frac{\pi}{2}-x))=-sin(\frac{\pi}{2}-x)=-cos x](https://tex.z-dn.net/?f=y%3D%20sin%28x-%5Cfrac%7B%5Cpi%7D%7B2%7D%29%3Dsin%28-%28%5Cfrac%7B%5Cpi%7D%7B2%7D-x%29%29%3D-sin%28%5Cfrac%7B%5Cpi%7D%7B2%7D-x%29%3D-cos%20x)
![sin(\frac{\pi}{2}-x)=cosx](https://tex.z-dn.net/?f=%20sin%28%5Cfrac%7B%5Cpi%7D%7B2%7D-x%29%3Dcosx%20)
We know that cos x is even function
Replace x by -x then, we get
....(cos (-x)=cos x)
Hence, the value of C=
for which given function will be an even function.
Answer:c.![\frac{\pi}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpi%7D%7B2%7D)